How Things Work: Mathematical Knots
A knot, in the conventional sense, is an intertwining rope or string usually designed to tie objects (or shoelaces) together. Take a piece of rope, twist, pull, and loop it a few times and you get a hitch knot. Loop it again and you get a half hitch knot. Loop it some more and you get a cow hitch knot. Glue those ends of the rope together, however, and you get a mathematical knot.
A knot, in the mathematical sense, is a conventional knot on a closed loop. Stated another way, mathematical knots lack loose ends. This is a critical, fundamental difference between mathematical knots and the knots we tie every day. Regular knots can be easily untangled by manipulating the loose ends of a rope; mathematical knots cannot. Given this definition, a plain rubber band can also qualify as a mathematical knot. This loop (a complete circle) is indeed a special kind of mathematical knot and is given its own name: the unknot.
Surely at this point, some of you are scratching your heads. If an unknot is just a loop, is it possible to untangle a mathematical knot into an unknot? Think about it. If a mathematical knot can be transformed into an unknot without cutting it up and gluing it back together, the knot never existed in the first place -- and knot theory is in trouble. If one can prove that it is impossible to transform a knot into an unknot, however, the existence of knots is proven -- and knot theory is saved.
Proving the existence of knots was a longstanding challenge in knot theory. In the 1930s, Kurt Reidemeister took a step in that direction by showing that all transformations between knots can be reduced to three basic moves. These moves (the twist, the cross, and the poke) are collectively referred to as the Reidemeister moves. Two mathematical knots are topologically equivalent if one knot can be transformed into the other by a finite series of twists, crosses, and pokes. If Reidemeister moves are incapable of transforming a knot into an unknot, the existence of knots is proven.
The proof itself is surprisingly intuitive, and can be followed by drawing a trefoil knot as a curve on a sheet of paper, with breaks where the knot crosses itself. It has three sections. These sections can be colored with three different colors so that at each crossing, the three sections involved have either the same color, or three different colors -- a property called tricolorability. Reidemeister moves preserve tricolorability, but the single-loop unknot has only one section and thus one color. So no amount of twisting and pulling can turn a tricolored knot into a single-colored unknot.
Why knots? The fascination with knots began in the 1800s, when scientists still believed that a luminiferous ether pervaded the universe. Lord Kelvin proposed that every element should have a distinct signature based on its entanglement with the ether, prompting mathematicians and scientists alike to conjure up pictures of knots. Although the theory about luminiferous ether was eventually disproved, mathematicians continued to pursue knot theory on purely abstract grounds for over a century.
In the 1980s, knot theory again found itself at the forefront of science -- this time in biology. DNA can be visualized as a convoluted knot that has been stretched, coiled and packed into the cell's nucleus. Topoisomerase enzymes must quickly untangle this knot to allow replication and transcription to occur. By modeling DNA as a closed loop, scientists were able to obtain a quantitative measure of DNA packing. Topology also allowed researchers to examine the enzyme's ability to untangle and tangle complicated knots in a quick and efficient manner.
Other esoteric and far-flung applications of knot theory are delightful to peruse at one's leisure. Interested readers can check out molecular knots and topological stereoisomers in Erica Flapan's When Topology Meets Chemistry. Physicists can flip through Dirk Kreimer's Knots and Feynman Diagrams. The ardent non-scientist is encouraged to take his shoelace and explore the 800 ways to tie it to his desk. I, for one, am profoundly impressed that this is an entire article on knots -- with knot one pun.